CS CS 175 – Week 4 Triangle Mesh Smoothing Discrete Differential Geometry

CS Overview triangle mesh basics data structures basic differential geometry discretization

CS Triangle Mesh Basics triangle mesh consist of vertices edges triangles and it is an orientable manifold (with boundary)

CS Triangle Mesh Basics terminology neighbours valency boundary Euler formula V – E + F = 2(1 – G) – B

CS Data Structures extended indexed face set array of vertices coordinates index to one adjacent triangle array of triangles vertex indices indices of neighbouring triangles

CS Data Structures directed half-edges array of vertices coordinates index to one adjacent half-edge array of half-edges vertex index neighbouring half-edge previous half-edge

CS Differential Geometry Definition of tangent plane surface normal first & second fundamental form curvature Laplace Beltrami operator

CS Discretization normal average adjacent triangle normals area or inverse area weighting average sphere normals …

CS Discretization integral curvature values blending cylinders Gaussian curvature Gauss-Bonnet-Theorem s K = 2  –  i  i mean curvature s H = ¼  i ||e i ||  i